복잡성 클래스 연산자에 대한 좋은 참조?

복잡성 클래스 연산자 에 대해 쓸 때 참조 할 수있는 좋은 설명 기사 나 설문 조사가 있는지에 관심 이 있습니다.

연산자의 예

다음은 답변이 설명 할 수있는 최소한의 연산자 목록으로 해석 될 수 있습니다. 여기서 $\mathbf C$ 는 임의의 유한 알파벳 대한 임의의 언어 세트입니다 $\Sigma$.

$\exists \mathbf C := \left\{ L \subseteq \Sigma^\ast \,\left|\, \begin{array}{l} \exists A \in \mathbf C \;\exists f \in O(\mathrm{poly}(n))\;\forall x \in \Sigma^\ast: \\\quad \bigl[x \in L \iff \exists c \in \Sigma^{f(|x|)}: (x,c) \in A \bigr] \end{array} \right\}\right.$

• $\exists$ 연산자 명백하게 표기 불구 바그너 [1]에 의해 도입 된 보다는 ∃ C . 이런 식으로 구성된 클래스의 가장 유명한 예는 N P = ∃ P 입니다. 이 연산자에는 상보 적 수량 자 ∀ 가 제공되는데,정의의 ∃ c 는 ∀ c 로 대체되어전체 다항식 계층 구조를 쉽게 정의 할 수 있습니다 (예 : Σ P 2 P = ∃ ∀ $\bigvee\! \mathbf C$$\exists \mathbf C$$\mathsf{NP} = \exists \mathsf P$$\forall$$\exists c$$\forall c$$\mathsf{\Sigma_2^P P = \exists \forall P}$ . 이것은 아마도 정의 된 첫 번째 연산자 일 수 있습니다.

$\oplus \mathbf C := \left\{ L \subseteq \Sigma^\ast \,\left|\, \begin{array}{l} \exists A \in \mathbf C \;\exists f \in O(\mathrm{poly}(n)) \;\forall x \in \Sigma^\ast: \\ \quad\bigl[x \in L \iff \#\{ c \in \Sigma^{f(|x|)}: (x,c) \in A \} \not\equiv 0 \pmod{2}\bigr]\end{array} \right\}\right.$

• 오퍼레이터는 비슷 ∃의 것을 오퍼레이터 ⊕ C는 클래스에서 검증되어 존재 인증서의 수에 관한 C를 대신 certficiates의 수는 모듈로 (modulo) 계산 2 . 이는 클래스를 정의하는 데 사용될 수 ⊕ P 및 ⊕ L이 . 다른 모듈러스 k 에 대해서도 유사한 연산자 " M o d k ⋅ "가 존재합니다 .$\oplus$$\exists$$\oplus\mathbf C$$\mathbf C$$2$$\mathsf{\oplus P}$$\mathsf{\oplus L}$$\mathsf{Mod_k \cdot}$$k$

$\mathsf{co}\mathbf C := \Bigl\{ L \subseteq \Sigma^\ast \,\Big|\, \exists A \in \mathbf C \;\forall x \in \Sigma^\ast: \bigl[x \in L \iff x \notin A \bigr] \Bigr\}$

• 이것은 상보 연산자이며 , c o C = P , c o M o d k L 및 보체로 닫히지 않은 것으로 알려진 클래스의 다른 클래스 를 정의하는 데 암묵적으로 사용됩니다 .$\mathsf{ coNP}$$\mathsf{coC_=P}$$\mathsf{coMod_kL}$

$\mathsf{BP}\cdot\mathbf C := \left\{ (\Pi_0, \Pi_1) \,\left|\, \begin{array}{l} \Pi_0, \Pi_1 \subseteq \Sigma^\ast \;\mathbin\&\; \exists A \in \mathbf C \; \exists f \in O(\mathrm{poly}(n))\;\forall x \in \Sigma^\ast: \\ \quad\left[\begin{array}{l} \bigl(x \in \Pi_0 \Leftrightarrow \#\{c \in \Sigma^{f(|x|)}: (x,c) \in A \} \leqslant \tfrac{1}{3} {|\Sigma^{f(|x|)}|} \bigr) \mathbin\& \\ \bigl(x \in \Pi_1 \Leftrightarrow \#\{c \in \Sigma^{f(|x|)}: (x,c) \in A \} \geqslant \tfrac{2}{3} {|\Sigma^{f(|x|)}|} \bigr) \end{array}\right]\end{array}\right\}\right.$

— with apologies for the spacing

• The $\mathsf{BP}$ operator was apparently introduced by Schöning [2], albeit to define languages (i.e. he did not permit a probability gap) and without using the explicit constants $\tfrac{1}{3}$ or $\frac{2}{3}$. The definition here yields promise-problems instead, with YES-instances $\Pi_1$ and NO-instances in $\Pi_0$. Note that $\mathsf{BPP = BP\cdot P}$, and $\mathsf{AM = BP\cdot NP}$; this operator was used by Toda and Ogiwara [3] to show that $\mathsf{P^{\#P} \subseteq BP\cdot\oplus P}$.

Remarks

Other important operators which one can abstract from the definitions of standard classes are $\mathsf{C_= \cdot\mathbf C}$ (from the classes $\mathsf{C_= P}$ and $\mathsf{C_= L}$) and $\mathsf C\cdot\mathbf C$ (from the classes $\mathsf{PP}$ and $\mathsf{PL}$). It is also implicit in most of the literature that $\mathsf{F\cdot}$ (yielding function problems from decision classes) and $\#\cdot$ (yielding counting classes from decision classes) are also complexity operators.

There is an article by Borchert and Silvestri [4] which propose to define an operator for each class, but which does not seem to be referred to much in the literature; I also worry that such a general approach may have subtle definitional issues. They in turn refer to a good presentation by Köbler, Schöning, and Torán [5], which however is now over 20 years old, and also seems to miss out $\oplus$.

Question

What book or article is a good reference for complexity class operators?

References

[1]: K. Wagner, The complexity of combinatorial problems with succinct input representations, Acta Inform. 23 (1986) 325–356.

[2]: U. Schöning, Probabilistic complexity classes and lowness, in Proc. 2nd IEEE Conference on Structure in Complexity Theory, 1987, pp. 2-8; also in J. Comput. System Sci., 39 (1989), pp. 84-100.

[3]: S. Toda and M. Ogiwara, Counting classes are at least as hard as the polynomial-time hierarchy, SIAM J. Comput. 21 (1992) 316–328.

[4]: B. and Borchert, R. Silvestri, Dot operators, Theoretical Computer Science Volume 262 (2001), 501–523.

[5]: J. Köbler, U. Schöning, and J. Torán, The Graph Isomorphism Problem: Its Structural Complexity, Birkhäuser, Basel (1993).

Here is a reference with many definitions of operators (not many details though):

S. Zachos and A. Pagourtzis, Combinatory Complexity: Operators on Complexity Classes, Proceedings of 4th Panhellenic Logic Symposium (PLS 2003), Thessaloniki, Jul 7-10 2003.

• It defines an identity operator $\mathcal E$, as well as operators $\mathsf{co}$- , $\mathsf{N}$ (corresponding to $\exists$ above), $\mathsf{BP}$, $\mathsf{R}$ (corresponding to bounded one-sided error), $\oplus$, $\mathsf U$ (corresponding to non-determinism with a unique accepting transition), $\mathsf P$ (corresponding to unbounded two-sided error), and $\Delta$ (which for a class $\mathbf C$ forms $\mathbf C \cap \mathsf{co}\mathbf C$).

• It shows that:

  1. $\mathcal E$ is an identity element with respect to composition [Definition 1];
  2. $\mathsf{co}$- is self-inverse [Definition 2];
  3. $\mathsf{N}$ is idempotent [Definition 3] — implicit is that $\mathsf{BP}$, $\mathsf{R}$, $\oplus$, $\mathsf U$, and $\mathsf{P}$ are also idempotent;
  4. $\mathsf{BP}$ and $\mathsf{P}$ commute with $\mathsf{co}$-  [Definitions 4 and 8], while $\oplus$ is invariant under right-composition with $\mathsf{co}$- [Definition 6];
  5. The above operators are all monotone (that is, ${\mathbf C_1 \subseteq \mathbf C_2} \implies {\mathcal O\cdot\mathbf C_1 \subseteq \mathcal O\cdot\mathbf C_2}$ for all operators $\mathcal O$ above):

Throughout, it also describes a handful of ways that these operators relate to traditional complexity classes, such as $\mathsf{\Sigma^p_2 P}$, $\mathsf{ZPP}$, $\mathsf{AM}$, $\mathsf{MA}$, etc.

As an introductory reference to the notion of a complexity operator (and demonstrating some applications of the idea), the best I have found so far is

D. Kozen, Theory of Computation (Springer 2006)

which is derived from lecture notes on computational complexity and related topics. On page 187 ("Supplementary Lecture G: Toda's Theorem"), he defines the operators

• $\mathsf R$ (for random certificates with bounded one-sided error, as in the class $\mathsf{RP}$)
• $\mathsf{BP}$ (for random certificates with bounded two-sided error, see above)
• $\mathsf P$ (for random certificates with unbounded error, c.f. $\mathsf C$ in the remarks above)
• $\oplus$ (for an odd number of certificates, see above)
• $\Sigma^{\mathrm{p}}$ (for existence of polynomial-length certificates, c.f. $\exists$ above)
• $\Sigma^{\mathrm{log}}$ (for existence of $O(\log n)$-length certificates, c.f. $\exists$ above)
• $\Pi^{\mathrm{p}}$ and $\Pi^{\mathrm{log}}$ (complementary operators to $\Sigma^{\mathrm{p}}$ and $\Sigma^{\mathrm{log}}$: see remarks on $\forall$ above)
• $\#$ (defining a counting class, c.f. remarks above)

and tacitly defines $\mathsf{co\text-}$ on page 12 in the usual way.

Kozen's treatment of these operators is enough to indicate how they are connected with the "usual" complexity classes, and to describe Toda's theorem, but does not much discuss their relationships and only mentions them for a total of 6 pages (in what is after all a book covering a much wider topic). Hopefully someone can provide a better reference than this.